Function Composition Exploration
Evaluate composite functions and enforce both domain restrictions. Includes a worked example, printable practice, and an answer key with explanations.
The answer key follows the practice below. In the editor, remove the final answer-key section before sharing a student copy.
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Function Composition Exploration
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Learning goal and worked example
Evaluate composite functions and enforce both domain restrictions.
If f(u) = 1/u and g(x) = x−3, then f(g(x)) = 1/(x−3), defined only for x ≠ 3. Always apply the inside function first.
Prerequisites: substitution, radicals, and interval notation. Composition usually depends on order.
Read each question carefully and provide clear, complete answers. Show all your work where applicable.
1. The notation (f ∘ g)(x) represents the of functions f and g.
2. To evaluate (f ∘ g)(x), you first apply the function to x, and then apply the function to the result.
3. The domain of (f ∘ g)(x) consists of all x in the domain of g such that g(x) is in the domain of .
Given f(x) = 2x + 3 and g(x) = x² - 1, find the following:
4. (f ∘ g)(2)
5. (g ∘ f)(-1)
6. (f ∘ f)(0)
Given f(x) = 3x - 2 and g(x) = x + 5, which of the following is (f ∘ g)(x)?
3x + 13
3x + 3
3x + 7
3x - 7
Given h(x) = x² and k(x) = √x, which of the following is (k ∘ h)(x)?
x
x²
|x|
√x
Given f(x) = 1/(x-2) and g(x) = √x, find the composite function and its domain.
7. Find (f ∘ g)(x).
8. State the domain of (f ∘ g)(x).
For any two functions f and g, (f ∘ g)(x) is always equal to (g ∘ f)(x).
True
False
If f(x) = x and g(x) = x², then (f ∘ g)(x) = x².
True
False
A store applies a 20% discount and a $10 coupon in either order. Let x be the original price in dollars and assume x ≥ 12.50 so neither calculation gives a negative price.
9. Write a function D(x) for the price after the 20% discount.
10. Write a function C(x) for the price after the $10 coupon.
11. Find (C ∘ D)(x) and explain what it represents in this context.
12. Find (D ∘ C)(x) and explain what it represents in this context.
13. Which order of applying the discounts results in a lower final price for an item that originally costs $100? Justify your answer.
Answer key and worked solutions
1. Composition. 2. g, then f. 3. f.
4. g(2)=3, so f(3)=9. 5. f(−1)=1, so g(1)=0. 6. f(0)=3, so f(3)=9.
Forming composite functions: first MCQ a, 3(x+5)−2=3x+13. Second MCQ c, √(x²)=|x| for all real x.
7. (f∘g)(x)=1/(√x−2). 8. Require x≥0 and √x≠2, so domain [0,4) ∪ (4,∞).
Properties: False (composition is not generally commutative); True (the identity function leaves x² unchanged).
9. D(x)=0.8x. 10. C(x)=x−10.
11. C(D(x))=0.8x−10: percentage discount first, coupon second. 12. D(C(x))=0.8(x−10)=0.8x−8: coupon first, percentage discount second.
13. At $100, percentage then coupon gives $70; coupon then percentage gives $72. Applying the percentage discount first saves $2 more.
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